# Relationship between continuous and discrete time models of a LTI system

Joined Oct 5, 2018
34
Hi everyone! I was recently posted a question by my professor
Discuss the underlying relationship between the continuous and discrete time models of a linear time invariant system. However, i am unsure of how to approach and answer the question. Any reference to models or understanding would be appreciated thank you!

My understanding so far is as such.
Discrete time signals are simply a collection of individual signals. These discrete signals can be a product of sampling a continuous time signal, or it can be a product of truly discrete phenomena. Thus, it can be stated that discrete time LTI is a sampling of continuous LTI. Additionally, it can also be stated from the aforementioned that it is also defined by differing time periods, i.e continuos is from t0 while sampling is from the sampling period taken ie t1 to t1+x.

I am unsure of how to proceed and any help will be appreciated thank you!

Joined Oct 5, 2018
34
Thank you so much! so is it fair for me to answer the question by saying that they share properties in the aspect of discrete LTI being a subset of continuous LTI?

I am quite unclear about what should be included in the relationship between the two to make for a concise answer.

Thank you again for your help!

#### MrAl

Joined Jun 17, 2014
9,633
Hi,

What you might do is start with some continuous system and then convert that to a sampled system and see what comes of it.
One thing you might find is that the sampled system may or may not mimic the continuous system at every point.
A good place to start is probably with some simple electrical system like maybe an RLC filter circuit, or maybe even just an RC low pass fitler circuit just to get a feel for what is happening. Convert the RC LP filter to a sampled system and notice what happens with the output with say a unit step input.
What we dont know is what your instructor usually accepts for answers. For example, what would he say if you just said:
"One system provides a solution for every possible point in time while the other just provides single point solutions in time spaced by the sampling interval".
That's probably not enough but only you could decide that, and if you give more it cant hurt.

Joined Oct 5, 2018
34
Hi,

What you might do is start with some continuous system and then convert that to a sampled system and see what comes of it.
One thing you might find is that the sampled system may or may not mimic the continuous system at every point.
A good place to start is probably with some simple electrical system like maybe an RLC filter circuit, or maybe even just an RC low pass fitler circuit just to get a feel for what is happening. Convert the RC LP filter to a sampled system and notice what happens with the output with say a unit step input.
What we dont know is what your instructor usually accepts for answers. For example, what would he say if you just said:
"One system provides a solution for every possible point in time while the other just provides single point solutions in time spaced by the sampling interval".
That's probably not enough but only you could decide that, and if you give more it cant hurt.
Precisely my fear Sir. I am not sure if my limited understanding would suffice in his question since it was too vague... But i thank you so much for your input. Maybe the inclusion of a case study will aid in the showcase of the concepts!

Thank you!

#### MrAl

Joined Jun 17, 2014
9,633
Precisely my fear Sir. I am not sure if my limited understanding would suffice in his question since it was too vague... But i thank you so much for your input. Maybe the inclusion of a case study will aid in the showcase of the concepts!

Thank you!
Hi,

Yeah i agree it is rather vague, so maybe the instructor just wants to get a feel for what you know about this, so mention all that you can think of. I remember though when i did this a while back the two graphs were informative because we could see where the sampled system was different than the continuous time system as to the actual results of the response of the system. There are so many ways to view this comparison though so maybe i'll look through some material later and see what other ideas i can find.
I am assuming that you can transform a continuous time system into some discrete system? If not we can go over that i guess as that will be only one part of your complete answer.
Another point is that the sample time affects the result accuracy. You could show some graphs maybe with different sample times.

Joined Oct 5, 2018
34
Hi,

Yeah i agree it is rather vague, so maybe the instructor just wants to get a feel for what you know about this, so mention all that you can think of. I remember though when i did this a while back the two graphs were informative because we could see where the sampled system was different than the continuous time system as to the actual results of the response of the system. There are so many ways to view this comparison though so maybe i'll look through some material later and see what other ideas i can find.
I am assuming that you can transform a continuous time system into some discrete system? If not we can go over that i guess as that will be only one part of your complete answer.
Another point is that the sample time affects the result accuracy. You could show some graphs maybe with different sample times.
ahh thank you for all your help with both posts! I will take into consideration your advice and put down the information necessary to ensure the answers are well substantiated.

Thank you so much again!

#### bogosort

Joined Sep 24, 2011
696
Hi everyone! I was recently posted a question by my professor
Discuss the underlying relationship between the continuous and discrete time models of a linear time invariant system. However, i am unsure of how to approach and answer the question. Any reference to models or understanding would be appreciated thank you!
Google Shannon sampling theorem. I'd also recommend reading section II of his original (and short) paper on the subject: "Communication in the presence of noise".